Group #141928 — subgroups

Subgroups — #141928

Order 512.

307 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 307 conjugacy classes of subgroups — too many to draw. Only its 94 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21736	C₂	1	1	yes	maximal, characteristic
2	#12882	C₂	1	1	yes	maximal, characteristic
3	#7844	C₂	1	1	yes	maximal, characteristic
4	#7182	C₂	1	2	yes	maximal
5	#7179	C₂	1	2	yes	maximal
6	#3162	C₄	1	1	yes	characteristic
7	#2961	—	2	1	no	—
8	C₂×#364	C₂²	1	2	yes	—
9	#1994	—	2	1	no	—
10	#1946	C₄	1	2	yes	—
11	#1945	—	2	1	no	—
12	#1849	C₄	1	1	yes	characteristic
13	#1773	—	2	1	no	—
14	#1772	C₂²	1	2	yes	—
15	#1744	—	2	1	no	—
16	C₄×#97	C₂²	1	1	yes	characteristic
17	C₂×#322	—	2	1	no	—
18	C₂×#322	C₂²	1	1	yes	characteristic
19	C₂×C₈²	C₂²	1	1	yes	characteristic
20	#1377	—	2	1	no	—
21	#1375	—	2	1	no	—
22	#533	—	2	1	no	—
23	C₂×M₃₂	—	2	2	no	—
24	#501	—	2	4	no	—
25	#446	—	4	4	no	—
26	C₂×#108	—	2	1	no	—
27	C₂×#108	C₂×C₄	1	2	yes	—
28	#427	—	2	1	no	—
29	#427	C₂×C₄	1	2	yes	—
30	#424	—	2	2	no	—
31	#423	—	2	1	no	—
32	#405	—	2	2	no	—
33	C₂×#97	C₂×C₄	1	1	yes	characteristic
34	C₂×#97	C₂×C₄	1	1	yes	characteristic
35	C₂×#97	D₄	1	1	yes	characteristic
36	C₂×#97	D₄	1	1	yes	characteristic
37	C₂×C₄×C₈	—	2	1	no	—
38	C₂×C₄×C₈	—	2	1	no	—
39	C₂×C₄×C₈	D₄	1	1	yes	characteristic
40	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
41	C₄³	D₄	1	1	yes	characteristic
42	#364	—	2	2	no	—
43	#364	—	2	2	no	—
44	#339	—	2	2	no	—
45	#339	—	2	2	no	—
46	#336	—	2	4	no	—
47	#322	—	2	4	no	—
48	#322	—	2	1	no	—
49	#322	Q₈	1	2	yes	complemented
50	C₈²	—	2	1	no	—
51	C₈²	Q₈	1	2	yes	complemented
52	C₂×M₁₆	—	2	2	no	—
53	C₂²×C₈	—	2	2	no	—
54	C₂²×C₈	—	2	2	no	—
55	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
56	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
57	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
59	C₄⋊Q₈	—	2	2	no	—
60	#125	—	2	2	no	—
61	C₂²⋊Q₈	—	4	2	no	—
62	C₄×Q₈	—	4	2	no	—
63	#117	—	2	2	no	—
64	C₂×C₄²	—	2	2	no	—
65	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
68	M₃₂	—	4	2	no	—
69	M₃₂	—	4	2	no	—
70	C₂×C₁₆	—	4	2	no	—
71	#108	—	2	4	no	—
72	#108	—	4	2	no	—
73	#107	—	2	2	no	—
74	#107	M₁₆	1	4	yes	—
75	#106	—	2	2	no	—
76	#106	—	2	2	no	—
77	C₄⋊C₈	—	4	2	no	—
78	C₄⋊C₈	—	4	4	no	—
79	#97	—	2	2	no	—
80	#97	—	2	2	no	—
81	#97	—	2	2	no	—
82	#97	—	2	2	no	—
83	#97	—	4	2	no	—
84	C₄×C₈	—	2	2	no	—
85	C₄×C₈	—	2	2	no	—
86	C₄×C₈	—	2	2	no	—
87	C₄×C₈	—	2	2	no	—
88	C₄×C₈	—	2	2	no	—
89	C₄×C₈	—	4	2	no	—
90	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
91	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
92	C₄×C₈	#41	1	1	yes	characteristic
93	C₄×C₈	#41	1	1	yes	characteristic
94	C₂×Q₈	—	4	2	no	—
95	C₂²×C₄	—	2	1	no	—
96	C₂²×C₄	—	2	1	no	—
97	C₂²×C₄	#104	1	2	yes	—
98	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
99	M₁₆	—	4	2	no	—
100	M₁₆	—	4	2	no	—
101	C₂×C₈	—	2	1	no	—
102	C₂×C₈	—	2	1	no	—
103	C₂×C₈	—	2	4	no	—
104	C₂×C₈	—	2	4	no	—
105	C₂×C₈	—	2	1	no	—
106	C₂×C₈	—	2	1	no	—
107	C₂×C₈	—	2	1	no	—
108	C₂×C₈	—	4	4	no	—
109	C₂×C₈	—	4	4	no	—
110	C₂×C₈	—	4	2	no	—
111	C₂×C₈	#104	1	2	yes	—
112	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
113	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
114	C₂×C₈	#124	1	1	yes	characteristic
115	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
116	C₂×C₈	C₂×M₁₆	1	2	yes	—
117	C₂×C₈	#131	1	2	yes	—
118	C₂×C₈	AGL(1,ℤ/8)	1	2	yes	—
119	C₂×C₈	#137	1	2	yes	—
120	C₄⋊C₄	—	2	4	no	—
121	C₄⋊C₄	—	4	4	no	—
122	C₄⋊C₄	—	4	2	no	—
123	C₂²⋊C₄	—	4	2	no	—
124	C₄²	—	2	1	no	—
125	C₄²	—	2	1	no	—
126	C₄²	—	2	1	no	—
127	C₄²	—	2	1	no	—
128	C₄²	—	2	1	no	—
129	C₄²	—	4	4	no	—
130	C₄²	—	4	2	no	—
131	C₄²	C₂²⋊Q₈	1	1	yes	characteristic
132	C₄²	C₂²⋊Q₈	1	1	yes	characteristic
133	C₁₆	—	8	2	no	—
134	C₂³	C₂×#104	1	1	yes	characteristic
135	Q₈	—	16	2	no	—
136	C₂×C₄	—	2	4	no	—
137	C₂×C₄	—	2	1	no	—
138	C₂×C₄	—	2	1	no	—
139	C₂×C₄	—	2	2	no	—
140	C₂×C₄	—	4	2	no	—
141	C₂×C₄	—	4	2	no	—
142	C₂×C₄	—	4	1	no	—
143	C₂×C₄	—	4	4	no	—
144	C₂×C₄	#391	1	1	yes	characteristic
145	C₂×C₄	#407	1	1	yes	characteristic
146	C₂×C₄	#408	1	1	yes	characteristic
147	C₂×C₄	#418	1	1	yes	characteristic
148	C₂×C₄	#419	1	1	yes	characteristic
149	C₂×C₄	C₂×#104	1	1	yes	characteristic
150	C₈	—	2	2	no	—
151	C₈	—	4	4	no	—
152	C₈	—	4	4	no	—
153	C₈	—	4	1	no	—
154	C₈	—	8	2	no	—
155	C₈	#446	1	2	yes	—
156	C₈	#446	1	2	yes	—
157	C₈	#489	1	2	yes	—
158	C₈	#501	1	2	yes	—
159	C₂²	—	2	2	no	—
160	C₂²	#1485	1	1	yes	characteristic
161	C₂²	#1490	1	1	yes	characteristic
162	C₂²	#1994	1	1	yes	characteristic
163	C₄	—	2	2	no	—
164	C₄	—	2	2	no	—
165	C₄	—	4	2	no	—
166	C₄	—	8	1	no	—
167	C₄	—	8	4	no	—
168	C₄	#1484	1	1	yes	characteristic
169	C₄	#1491	1	1	yes	characteristic
170	C₄	#1985	1	1	yes	characteristic
171	C₄	#1993	1	1	yes	characteristic
172	C₂	—	2	1	no	—
173	C₂	—	2	1	no	—
174	C₂	#9877	1	1	yes	characteristic
175	C₂	#11670	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21736

C₂

yes

maximal, characteristic

#12882

C₂

yes

maximal, characteristic

#7844

C₂

yes

maximal, characteristic

#7182

C₂

yes

maximal

#7179

C₂

yes

maximal

#3162

C₄

yes

characteristic

#2961

—

C₂×#364

C₂²

yes

—

#1994

—

#1946

C₄

yes

—

#1945

—

#1849

C₄

yes

characteristic

#1773

—

#1772

C₂²

yes

—

#1744

—

C₄×#97

C₂²

yes

characteristic

C₂×#322

—

C₂×#322

C₂²

yes

characteristic

C₂×C₈²

C₂²

yes

—

—

—

C₂×M₃₂

—

—

—

—

C₂×C₄

yes

—

#427

—

#427

C₂×C₄

yes

—

#424

—

#423

—

#405

—

C₂×#97

C₂×C₄

yes

characteristic

C₂×#97

C₂×C₄

yes

characteristic

C₂×#97

D₄

yes

characteristic

C₂×#97

D₄

yes

characteristic

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

C₂³

yes

characteristic

C₄³

D₄

yes

—

—

—

—

—

—

—

Q₈

yes

complemented

C₈²

—

C₈²

Q₈

yes

complemented

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₂²⋊C₄

yes

characteristic

C₂²×C₈

C₂²⋊C₄

yes

characteristic

C₂²×C₈

C₂²×C₄

yes

characteristic

C₂²×C₈

C₂×D₄

yes

characteristic

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

—

C₂×C₄²

C₂²⋊C₄

yes

characteristic

C₂×C₄²

C₂²⋊C₄

yes

characteristic

C₂×C₄²

C₂×D₄

yes

characteristic

M₃₂

—

M₃₂

—

C₂×C₁₆

—

—

—

—

M₁₆

yes

—

#106

—

#106

—

C₄⋊C₈

—

C₄⋊C₈

—

—

—

—

—

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×Q₈

yes

characteristic

C₄×C₈

C₂×Q₈

yes

C₄×C₈

yes

C₄×C₈

yes

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

#104

yes

—

C₂²×C₄

C₂×C₂²⋊C₄

yes

characteristic

M₁₆

—

100

M₁₆

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

#104

yes

—

112

C₂×C₈

C₄×Q₈

yes

characteristic

113

C₂×C₈

C₄×Q₈

yes

characteristic

114

C₂×C₈

#124

yes

characteristic

115

C₂×C₈

C₄⋊Q₈

yes

characteristic

116

C₂×C₈

C₂×M₁₆

yes

—

117

C₂×C₈

#131

yes

—

118

C₂×C₈

AGL(1,ℤ/8)

yes

—

119

C₂×C₈

#137

yes

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₂²⋊C₄

—

124

C₄²

—

125

C₄²

—

126

C₄²

—

127

C₄²

—

128

C₄²

—

129

C₄²

—

130

C₄²

—

131

C₄²

C₂²⋊Q₈

yes

characteristic

132

C₄²

C₂²⋊Q₈

yes

characteristic

133

C₁₆

—

134

C₂³

C₂×#104

yes

characteristic

135

Q₈

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

#391

yes

characteristic

145

C₂×C₄

#407

yes

characteristic

146

C₂×C₄

#408

yes

characteristic

147

C₂×C₄

#418

yes

characteristic

148

C₂×C₄

#419

yes

characteristic

149

C₂×C₄

C₂×#104

yes

characteristic

150

C₈

—

151

C₈

—

152

C₈

—

153

C₈

—

154

C₈

—

155

C₈

#446

yes

—

156

C₈

#446

yes

—

157

C₈

#489

yes

—

158

C₈

#501

yes

—

159

C₂²

—

160

C₂²

#1485

yes

characteristic

161

C₂²

#1490

yes

characteristic

162

C₂²

#1994

yes

characteristic

163

C₄

—

164

C₄

—

165

C₄

—

166

C₄

—

167

C₄

—

168

C₄

#1484

yes

characteristic

169

C₄

#1491

yes

characteristic

170

C₄

#1985

yes

characteristic

171

C₄

#1993

yes

characteristic

172

C₂

—

173

C₂

—

174

C₂

#9877

yes

characteristic

175

C₂

#11670

yes

—

	subgroup	G/H
center	C₂×C₄	#418
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1490
prime radical	C₂³	C₂×#104
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₄×#97 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₂×C₄

#418

commutator

C₂²×C₈

C₂²×C₄

Frattini, cosocle

C₂×C₄×C₈

C₂³

C₂²

C₂³

G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₄×#97 ◁ G

LECS

G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
49. #322	Q₈	semidirect
51. C₈²	Q₈	semidirect

H ≅ G/N

type

49. #322

Q₈

semidirect

51. C₈²

Q₈

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#21736	2
#12882	2
#7844	2
#7182	2
#7179	2

subgroup

index

Subgroups — #141928

Conjugacy classes of subgroups

Complements

Maximal subgroups